Function-defines-curvature curves

These images are fun pretty images made by defining a function for the curvature of the shape and then drawing forward and/or backwards.

The curves that jump off a smooth path are I guess a result of computational inaccuracies, but I haven't confirmed that. They make pretty shapes either way.

These are drawn with GNU Octave. The code to produce these is very simple, see the end of the page. TODO: add the codes here for each.

One example with numerical integration: (change the function and the drawing interval and step to make your own curves). These run in GUN Octave or MATLAB.

t = -10:0.01:10; %väli
kaarevuus =t.^2 -2.19;
x = cumtrapz(cos(cumtrapz(kaarevuus)));
y = cumtrapz(sin(cumtrapz(kaarevuus)));
plot(x,y)

Or alternatively with symbolic integration:

pkg load symbolic %octave tarvii tän
v = -20:0.1:20; %väli

%käytetään symbolista integrointia
syms t;
kaarevuusfunktio = symfun(t-2,t); %sym
integraalifunktio = int(kaarevuusfunktio,t); %sym  

kosinifunktio = cos(integraalifunktio);
xfunktio = int(kosinifunktio);
xx =matlabFunction(xfunktio); %this breaks on octave with complicated integrations, matlab can handle more

sinifunktio = sin(integraalifunktio);
yfunktio = int(sinifunktio);
yy =matlabFunction(yfunktio);

x=xx(v);
y=yy(v);
plot(x,y)